After few technical steps that can be found in [1]}, we can obtain the multipole expanded source action, first introduced in [2]}, and then in [1]}:
\(S_{source}&=&\frac{1}{2}\int dt[\ E\bar{h}_{00}+\mathbf {P}^i\bar{h}_{0i}-E\mathbf {Z}^i\partial _i\bar{h}_{00}-\mathbf {L}^i\epsilon _{ijk}\partial _j\bar{h}_{0k}] \nonumber \\& & -\int dt \sum _{l=2}^{\infty }\frac{1}{l!}I^L\partial _{L-2}\mathcal {E}_{k_{l-1}k_l}-\int dt \sum _{l=2}^{\infty }\frac{2l}{(l+1)!}J^L\partial _{L-2}B_{k_{l-1}k_l}\) 
